**1. Introduction**

Nanofluids, including nanomaterial dispersed in a pure fluid, are becoming applicable fluids in various systems due to their proved superior specification [1]. Augmented thermal conductivity is a remarkable property induced from nanofluid as compared with conventional fluids [2]. On the other hand, the viscosity of nanofluids is significantly varied depending on the type of nanoparticles, base fluid, and their interaction [3]. Some authors have observed Newtonian behaviour of nanofluids, while a non-Newtonian one has been widely revealed [4]. The non-Newtonian behaviours have practical implementations in wire and blade coating, molten plastic, dyeing of textile, some petroleum fluids, biological fluids movement, and food and slurries processing. In this regard, various kinds of rheological behaviours can be expected, defined by models such as power law, micropolar,

**Citation:** Abu-Hamdeh, N.H.; Aljinaidi, A.A.; Eltaher, M.A.; Almitani, K.H.; Alnefaie, K.A.; Abusorrah, A.M.; Safaei, M.R. Implicit Finite Difference Simulation of Prandtl-Eyring Nanofluid over a Flat Plate with Variable Thermal Conductivity: A Tiwari and Das Model. *Mathematics* **2021**, *9*, 3153. https://doi.org/10.3390/math9243153

Academic Editor: Mostafa Safdari Shadloo

Received: 1 October 2021 Accepted: 1 December 2021 Published: 7 December 2021

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Reiner–Philippoff, viscoelastic, Casson, Carreau, Giesekus, Prandtl, Prandtl–Eyring, and Powell–Eyring [5]. These models introduce special impendence on the momentum conservation equations to be compatible with the targeted behaviour. Indeed, in mathematical language, the relationship between shear stress and deformation rate is described by each model. Prandtl and Prandtl–Eyring are a function of sine inverse and sine hyperbolic, respectively [6,7]. The power law model characterizes the relation as nonlinear [8]. Sajid et al. [9] studied a micropolar Prandtl fluid for a porous stretching sheet situation. They assumed that the heat source is related to temperature and a chemical reaction occurs into the medium. Maleki et al. [10] performed numerical research on power law nanofluid, which flows on a porous plate. They found that using Newtonian nanofluid has no improvement effect on heat transfer.

In contrast, the non-Newtonian one had an essential role in boosting heat transfer. Shankar and Naduvinamani [11] worked on the transport phenomena of a Prandtl–Eyring fluid through a sensor surface under magnetic force conditions. The observation showed that magnetic parameter augmentation causes a velocity field increment and temperature profile reduction. At the same time, heat transfer diminished by the Prandtl number rose. Temperature and concentration variations on Prandtl–Eyring fluid heat transfer were investigated by Al-Kaabi and Al-Khafajy [12] in a porous medium. Finally, Hayat et al. [13] evaluated the efficacy of Prandtl–Eyring nanofluid on gyrotactic microorganisms in a stretching sheet. The results indicated that higher melting parameters hike the velocity and pull down the temperature.

Stretching surface is a well-known and habitual process in industrial situations, i.e., extrusion, fiberglass, cooling of the metallic plate, glass blowing, hot rolling, etc. Boundary layer flow and heat transfer is the theory that helps better understand the scientific phenomena underlying it [14]. Nonlinear equations are expected from practical problems which are experienced in engineering applications. The Keller box method is an implicit finite difference method used to solve these types of equations [15]. Munjam et al. [16] proposed a new technique to solve the fluid flow of a Prandtl–Eyring fluid on a stretched sheet and compared their results with the Keller box method. The analytical outcomes indicated that as the fluid parameter rises, velocity enhances. In addition, they found that the Prandtl–Eyring fluid induces a grosser velocity value as compared to the viscous one.

Jamshed et al. [17] explored the entropy generation of Casson nanofluid by considering the Tiwari and Das model and the Keller box method to solve ODEs. Two methanol-based nanofluids were used by introducing Cu and TiO2 nanoparticles; Cu nanofluids showed a better performance. In [18], they also used the same models and techniques for the same nanoparticles for engine oil base fluid. They concluded that entropy generation would enhance by Reynolds number and Brinkman numbers. Moreover, increasing nanofluid concentration led to shear rate enhancement. Abdelmalek et al. [19] discussed a Prandtl– Eyring nanofluid which influences Brownian motion and thermophoretic force through a stretched surface. It was proved that magnetic force is undesirable for the momentum. In contrast, Brownian motion and thermophoretic force raised the thermal energy.

In the viewpoint of heat transfer, thermal conductivity is a determinant parameter that is generally assumed to be unchanged. However, extensive studies emphasized that the efficacy of temperature changing the thermal conductivity would vary. Particularly, nanofluids have an intimate relation with temperature, which can considerably affect the heat transfer due to the higher aspect ratio that nanoparticles provide within the base fluid. It is proved that at higher temperatures, the thermal conductivity is typically more elevated. Thus, in a range of temperatures, the thermal conductivity is variable. Maleki et al. [20] studied the efficacy of different kinds of nanofluids on the heat transfer of a porous system. They claimed that the results are opposite with other researchers, i.e., adding more nanoparticles dwindled the heat transfer because it can alter radiation, viscous dissipation, and heat generation. In [21], they also surveyed the non-Newtonian nanofluids by considering the mixture of CMC and water as a base fluid. It was revealed that using non-Newtonian nanofluid in injection mode has a higher heat transfer efficiency as compared to the Newtonian one. Jamshed et al. [22] investigated Casson nanofluid in a stretching sheet system that included variable thermal conductivity. Keller box was the technique that solved ODEs. In this method, differential equations are solved numerically to reduce them into the 1st order differential equations. They used TiO2 and Cu as nanoparticles in water. Cu/water nanofluid had better heat conduction performance. Carreau–Yasuda nanofluid was researched by Waqas et al. [5] by considering gyrotactic motile microorganisms. Velocity, thermal, and temperature fields were amended by decreasing the bioconvection Rayleigh number, increasing the thermal Biot number, and decreasing the Prandtl number. In addition, the concentration field improved by reducing Brownian motion. Xiong et al. [23] explained that variable thermal conductivity has a determinant role in field quantities. They scrutinized a fibre-reinforced generalized thermoelasticity system by considering temperature-dependent thermal conductivity. Ibrahim and Negera [24] inspected an MHD Williamson nanofluid effect within a stretching cylinder by considering chemical reaction conditions. They asserted that the higher the parameter of variable thermal conductivity, the higher the Sherwood number and skin friction, while the lower the Nusselt number. Dada and Onwubuoya [25] analysed an MHD Williamson fluid over a stretchable surface of variable thickness and thermal conductivity. The conclusion indicated that rising changeable thermal conductivity improves temperature. Hasona et al. [26] described the variable thermal conductivity of a non-Newtonian nanofluid in a special geometry channel. They reported that rising thermal and electrical conductivities enhance the temperature of working fluid, which in turn augments heat transfer performance within the system. Fatunmbi and Okoya [27] presented an investigation on hydromagnetic Casson nanofluid at the attendance of thermophoresis, ohmic heating, and a nonuniform heat source with variable thermal conductivity for a stretching sheet system. They demonstrated that driving up the Casson fluid parameter dwindled the fluid flow velocity, albeit, it augmented the viscous drag.

After a glance into the erstwhile studies, since most industrial fluids include non-Newtonian fluids in a situation like stretching surface, the significant concerns of the current project are to discuss the Prandtl–Eyring nanofluid flow over a stretching sheet under three conditions of variable thermal conductivity, thermal radiative flow, and porous material. In addition, their effects on the entropy formation were elaborated by considering the Tiwari and Das model. In this way, Al2O3/EO and Cu/EO nanofluids were analysed at volume concentrations of 3% to 20%. Furthermore, this study implemented the implicit finite difference method to solve the boundary layer equations nicely. Therefore, it can be said that the valuable outcomes of this research can be a guideline for practical applications because it was conducted to select parameters close to actual industrial conditions.
